Click
Here
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, C06006, doi:10.1029/2009JC005821, 2010
for
Full
Article
Isopycnal diffusivities in the Antarctic Circumpolar Current
inferred from Lagrangian floats in an eddying model
A. Griesel,1 S. T. Gille,1 J. Sprintall,1 J. L. McClean,1 J. H. LaCasce,2 and M. E. Maltrud3
Received 23 September 2009; revised 9 December 2009; accepted 5 January 2010; published 10 June 2010.
[1] Lagrangian subsurface isopycnal eddy diffusivities are calculated from numerical
floats released in several regions of the Antarctic Circumpolar Current (ACC) of the 0.1°
Parallel Ocean Program. Lagrangian diffusivities are horizontally highly variable with
no consistent latitudinal dependence. Elevated values are found in some areas in
the core of the ACC, near topographic features, and close to the Brazil‐Malvinas
Confluence Zone and Agulhas Retroflection. Cross‐stream eddy diffusivities are depth
invariant in the model ACC. An increase of Lagrangian eddy length scales with depth is
masked by the strong decrease with depth of eddy velocities. The cross‐stream diffusivities
average 750 ± 250 m2 s−1 around the Polar Frontal Zone. The results imply that
parameterizations that (only) use eddy kinetic energy to parameterize the diffusivities are
incomplete. We suggest that dominant correlations of Lagrangian eddy diffusivities with
eddy kinetic energy found in previous studies may have been due to the use of too
short time lags in the integration of the velocity autocovariance used to infer the
diffusivities. We find evidence that strong mean flow inhibits cross‐stream mixing within
the ACC, but there are also areas where cross‐stream diffusivities are large in spite of
strong mean flows, for example, in regions close to topographic obstacles such as the
Kerguelen Plateau.
Citation: Griesel, A., S. T. Gille, J. Sprintall, J. L. McClean, J. H. LaCasce, and M. E. Maltrud (2010), Isopycnal diffusivities in
the Antarctic Circumpolar Current inferred from Lagrangian floats in an eddying model, J. Geophys. Res., 115, C06006,
doi:10.1029/2009JC005821.
1. Introduction
[2] Mixing generated by mesoscale eddies is believed to
play an important role in the transfer of water masses and
tracers across the Antarctic Circumpolar Current (ACC)
[Johnson and Bryden, 1989; Marshall et al., 1993; Döös
and Webb, 1994; Speer et al., 2000; Bryden and
Cunningham, 2003; Marshall and Radko, 2003]. This
eddy “mixing” is thought to be highly variable in time and
space [Visbeck et al., 1997; Treguier et al., 1997; Stammer,
1998]. In coarse resolution ocean models, an eddy diffusivity parameterization is commonly used to incorporate the
effects of eddies [Gent and McWilliams, 1990; Griffies,
1998]. However, it remains unclear whether eddy diffusion represents an appropriate parameterization. In addition,
coarse resolution models are sensitive to the magnitude, and
the horizontal and vertical distributions of the eddy mixing
coefficients [Danabasoglu and Marshall, 2007].
[3] Lagrangian floats provide a way to test the applicability of the eddy diffusion model (see LaCasce [2008] for
1
Scripps Institution of Oceanography, University of California, San
Diego, La Jolla, California, USA.
2
Meteorology and Oceanography Section, Department of Geosciences,
University of Oslo, Oslo, Norway.
3
Los Alamos National Laboratory, Los Alamos, New Mexico, USA.
Copyright 2010 by the American Geophysical Union.
0148‐0227/10/2009JC005821
an overview). Taylor [1921] showed how Lagrangian particles in isotropic and homogeneous turbulence spread diffusively, with a constant diffusivity calculated from the
integral of the velocity autocorrelation. Davis [1987, 1991]
refined the theoretical framework to allow both the mean
flow and diffusivity to be spatially variable, and to allow the
theory to be applied to ocean float data. In an ocean with
inhomogeneous, strong mean flows and with an eddy field
of varying horizontal scale, one challenge is to find the
asymptotic behavior of the velocity autocorrelation, which
requires sufficiently long time lags [e.g., Bauer et al., 1998;
Veneziani et al., 2004; Davis, 2005]. A second challenge
arises because of the presence of vortices or meanders that
lead to oscillations in the autocovariances. The oscillations
cause regimes of primarily subdiffusive behavior at intermediate time lags [Berloff and McWilliams, 2002a, 2002b;
Veneziani et al., 2005b]. Questions remain as to whether this
is a transient behavior, and whether the diffusive limit can
still be reached after long enough time lags.
[4] Diffusivities are hypothesized to be connected to the
kinetic energy of the mixing eddies. For example, both
Keffer and Holloway [1988] and Stammer [1998] used altimetric sea surface height variability to estimate surface eddy
diffusivities. Diffusivities inferred from Lagrangian floats
are also often highly correlated with eddy kinetic energy
[Krauss and Böning, 1987; Lumpkin et al., 2001; Zhurbas
and Oh, 2004; Sallée et al., 2008]. A recent study found
C06006
1 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
variations in eddy stirring and mixing rates using finite time
Lyapunov exponents to be closely related to variations in
eddy kinetic energy [Waugh and Abraham, 2008]. While
eddy activity is enhanced in regions with strong currents
such as in the ACC, these strong currents can also act as
mixing barriers [Bower et al., 1985], leading to small
effective eddy diffusivities [e.g., Marshall et al., 2006; A. C.
Naveira Garabato et al., Eddy‐induced mixing in the Southern Ocean, submitted to Journal of Physical Oceanography,
2009]. At any given geographic location, the magnitude of
the diffusivity depends on the relative importance of these
two factors [Shuckburgh et al., 2009a, 2009b]. For the near‐
surface Southern Ocean, Sallée et al. [2008] found a local
maximum of Lagrangian diffusivities in the core of the ACC
that, at some longitudes, seemed to be correlated with
enhanced eddy kinetic energy. In contrast, using a different
method, Marshall et al. [2006] obtained effective diffusivities at the surface about half as large as Sallée et al. [2008],
with a minimum of diffusivities in the core of the ACC jet.
This implied that the ACC jet acted as a mixing barrier in
the along‐streamline average.
[5] Inferences of depth dependence of eddy diffusivities
often rely on whether eddy kinetic energy or the eddy length
scales are used to determine the diffusivity. If eddy diffusivity scales like the eddy heat flux or like eddy kinetic
energy, a decrease with depth might be expected [Phillips
and Rintoul, 2000]. Such a decrease was found for the
Southern Ocean by Ferreira et al. [2005], who fitted eddy
stresses in a coarse model to hydrographic data; by Olbers
and Visbeck [2005] from a simple inverse approach using
hydrographic and wind stress data; and by Eden [2006] and
Griesel et al. [2009] using methods based on flux‐gradient
relationships. Conversely, critical layer theory predicts that
the diffusivity should have a maximum at the depth where
the speed of the mean flow is comparable to the phase speed
of Rossby waves [Green, 1970; Killworth, 1997; Smith and
Marshall, 2009]. Critical layer theory suggests that at the
ocean surface, mixing efficiency is reduced since the strong
current carries the eddies downstream so quickly that they
do not have much time to mix. However, at a depth of about
1–2 km, the eddies propagate at the same speed as the mean
flow, and eddies are therefore able to mix fluid elements.
This leads to higher mixing rates at depth. Indeed, enhanced
diffusivities at depth in the ACC have been diagnosed by
Treguier [1999], Cerovecki et al. [2009], Smith and
Marshall [2009], and Abernathy et al. [2009] in models of
varying complexity, using either flux‐gradient relationships
or the Nakamura [1996] method that evaluates the stretching
and folding of the contours of a passive tracer as it is mixed
and stirred by the eddies. At present, virtually no reliable
observations exist in the Southern Ocean to validate the
depth dependence of the model eddy diffusion coefficients.
[6] In this paper, we estimate isopycnal eddy diffusivities
from the dispersion of Lagrangian numerical floats released
in core regions of the ACC in a fine resolution configuration
of the Parallel Ocean Program (POP). Our intent is first to
explore whether a diffusive limit can be reached after long
enough time lags of the velocity autocorrelations. This effort
will demonstrate that the theoretical framework of Taylor
[1921] and Davis [1987, 1991] is appropriate for determining representative diffusivities if appropriate means are
subtracted. Using these eddy diffusivities, our second goal is
C06006
to determine the magnitude, and the horizontal and vertical
distributions of the Lagrangian diffusivities in the Southern
Ocean. Our study will show elevated diffusivities in the core
of the ACC and in eddy‐rich areas such as the Malvinas
Confluence and the Agulhas Retroflection. However, we
find the cross‐stream eddy diffusivities to have no significant depth dependence. Finally, our third objective is to
examine the relationship of the diffusivities to the eddy
kinetic energy and strength of the mean flow. Our results
suggest that parameterizations that only use eddy kinetic
energy to parameterize the diffusivities are misleading. We
also find that Lagrangian diffusivities are not consistently
reduced in regions with strong mean flow.
[7] This paper is organized as follows. In section 2 we
will introduce the model and outline the float deployments
and trajectories. Section 3 discusses the method used to
compute the Lagrangian diffusivities. Section 4 examines
whether a diffusive limit is reached, and what role the mean
flow, streamline projections and rotational components of
the eddies play. Section 5 shows the horizontal and vertical
distributions of the binned Lagrangian diffusivities and
discusses their relationship to the eddy kinetic energy and
mean flow. Section 6 presents the conclusions.
2. Model Description
[8] We use the 1/10° global POP model, described in
detail by Maltrud and McClean [2005] and used by Griesel
et al. [2009] to assess eddy heat flux parameterization in the
Southern Ocean. This simulation has one of the highest
horizontal resolutions used to date in global ocean modeling. Its Mercator grid results in a horizontal resolution
between 4 and 9 km in the Southern Ocean, which is sufficient to resolve the first baroclinic Rossby radius north of
about 50°S. Vertical resolution is between 10 m at the
surface and 250 m at depth. The model was spun up for
14 years (1979–1993), and then run with realistic forcing
from 1994 to 2003 (see Maltrud and McClean [2005] for
details). Lagrangian eddy scales calculated from surface
drifters in the model have been shown to be comparable to
observations in the North Atlantic [McClean et al., 2002].
[9] Horizontal diffusion is biharmonic for tracers and
momentum, as discussed by Maltrud and McClean [2005]
and by Griesel et al. [2009]. For vertical mixing in the
model, the K profile parameterization of Large et al. [1994]
is used, with background mixing coefficients ranging from
0.1 × 10−4 m2s−1 near the surface to 1 × 10−4 m2s−1 at depth.
The subgrid scale parameterizations, as well as spurious
numerical diffusion originating from the tracer advection
discretization can all potentially introduce diapycnal velocities through the Veronis [1975] effect, and thus may contribute to the Lagrangian motions of particles in the model.
For the purpose of this paper we assume this effect to be
negligible.
2.1. Float Trajectories
[10] Our analysis focuses on numerical floats deployed in
4 patches of 80 floats at each of three depths, initially at
300 m, 800 m, and 1500 m (Figure 1). The deployment
patches were sited within the core of the ACC around 55°S.
In each patch, floats were deployed on a grid with quarter
degree spacing in latitude and longitude. This spacing was
2 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
Figure 1. (left) Trajectories after floats have traveled for 3 years (1045 days) in 1998–2000 (Period 1),
and deployment locations for the four patches (P1–P4) at deployment depths (a) 300, (b) 800, and
(c) 1500 m. (right) Trajectories for days 1046–1761 in 1999–2000 (Period 2), and deployment locations
for the four patches (P1–P4) at deployment depths (d) 300, (e) 800, and (f) 1500 m. Color is float depth.
Also shown are geostrophic streamlines Yg = gf−1h that enclose the approximate location of the ACC.
Streamlines were computed using the model’s 1998–1999 mean sea surface height h and filtered with
a 51‐point triangular filter (equivalent to 5°). Streamlines indicate from north to south the approximate
mean location of Subtropical Front, Subantarctic Front, Polar Front, and Southern ACC Front.
3 of 18
C06006
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
C06006
Figure 2. Potential density anomaly Ds (kg m−3) for each float point as a function of depth for the first
1045 days (Period 1) using all deployments. Also indicated are the mean potential density and standard
deviation for each deployment. Patch numbers 1–4 and deployment depths 300, 800, and 1500 m are
shown. Red dots indicate floats have entered the mixed layer.
chosen to allow the floats to move somewhat independently.
Closer spacing was judged unlikely to yield additional
information, because submesoscale processes are not
resolved in the model. For the same reason, high‐frequency
dispersion statistics might not be expected to be fully representative of the ocean. A first group was deployed on
1 January 1998 and a second group at the same locations
60 days later on 2 March 1998. Here, floats for the two
deployments will be combined, since statistically consistent
convergence properties, error bars and diffusivities are
found from each deployment. After deployment, the floats
were advected by the three‐dimensional model flow. Trajectory positions were computed using a predictor‐corrector
scheme with a time step of 6.3 min corresponding to the
tracer time step. At midnight each day, temperature, salinity
and velocities were interpolated to the current trajectory
positions and saved.
[11] The first group of floats traveled for a total of 1761
days (almost 5 years), and the second group traveled for
1701 days. After day 1045 (almost 3 years), the model was
restarted at day 1 January 1999. The final locations from the
first 1045 days were used as initial conditions for the next
716 days (almost 2 years) for both groups. The
reinitialization leads to a discontinuity in the temperature
and salinity fields and a less significant one in the velocity
fields. Therefore, we treat the trajectories from days 1–1045
and those from day 1046–1761 separately and denote them
Period 1 and Period 2 respectively. Figures 1a–1c show the
trajectories of all floats for the three deployment depths until
day 1045 (Period 1). Figures 1d–1f show the trajectories for
days 1045 to 1761 (Period 2). During the first three years
most floats stay within the boundaries of the ACC
(Figures 1a–1c). During Period 2 a few floats deployed at
300 m upwell and cross the ACC at the surface (Figure 1d).
2.2. How Isopycnal Are the Floats?
[12] If the floats approximately follow isopycnal surfaces
they travel to deeper depths when they are north of their
deployment location and to shallower depths when they are
to the south (Figure 1). Figure 2 shows potential density
anomalies as a function of float depth, with respect to the
initial potential density for each float, for all patches and
deployment depths. Only Period 1 floats are considered,
since, as mentioned, the reinitialization led to a discontinuity
in the density fields. The labels in Figure 2 indicate the
mean potential density (averaged over all floats over the
1045 float days) and the standard deviation. The standard
deviation reflects three factors. Some of the variability
(O(10−2kg m−3)) can be explained by the quarter degree
spacing used for the float deployment grid, which means
that the floats do not all start at precisely the same density.
The different initial conditions in the two groups that were
4 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
deployed 60 days apart contribute very little to the standard
deviation. The largest part of the standard deviation is due to
the fact that potential density is not perfectly conserved
along the individual float trajectories. However, the individual floats in the 800 m deployments experience anomalies smaller than 0.2 kg m−3 and the 1500 m deployments
experience anomalies smaller than 0.1 kg m−3. The trajectories of the 800 and 1500 m deployments in Patches 1 and 2
are more constrained by topography and experience less
density change than those in Patches 3 and 4. Patch 2 floats,
for example, have to travel through a narrow strait around
Kerguelen Plateau, whereas the floats from Patch 3 travel
more freely across the abyssal plains west of Drake Passage.
[13] Each float experiences density changes once it
encounters the surface mixed layer (indicated in red in
Figure 2). Here, the base of the mixed layer is defined as the
depth at which the monthly mean potential density differs
from surface density by 0.03 kg m−3. Mixed layer
encounters occur for 15–30% of the float days for the 300 m
deployments, 1–4% for the 800 m deployments, and less
than 0.5% for the 1500 m deployments. In this study, we
exclude all data from a given float after it encounters the
surface or the mixed layer. Away from the mixed layer,
density is changed through diapycnal mixing arising from
explicit and implicit diapycnal diffusion, and from the
resolved diapycnal eddy velocities. A detailed investigation
about the origin of the density anomalies requires the use of
additional data and must be left to future work.
3. Method
[14] Davis [1991] showed how the evolution of the mean
tracer concentration in inhomogeneous flows with finite
scale eddies depends only on the initial condition of the
mean and source fields and the single particle statistics (see
Appendix A). For times longer than the typical eddy scales,
the eddy flux at time t and location x can then be approximated by a typical flux‐gradient relationship with a diffusivity ∞. This diffusivity is determined by the integral of
the Lagrangian autocovariance function
Z
ij ðx; Þ ¼
0
d ~hu0 i ðt0 jx; t0 Þ u0 j ðt0 þ ~jx; t0 ÞiL ;
1
ij ¼ lim!1 ij ðx; Þ;
ð1Þ
ð2Þ
where u′i(t0 + t∣x, t0) denotes the residual velocity of a
particle at time t0 + t passing through x at time t0 (see
Appendix A). RL = hu′i (t0∣x, t0) u′j (t0 + ~∣x, t0)iL is the
Lagrangian autocovariance, and h iL is the Lagrangian
average over many trajectories (Figure A1). For stationary
and homogeneous statistics, the diffusivity is equal to half
the time rate of change of the dispersion hr′2i
1
ij ðx; Þ ¼ dt hri0 ðjx; t0 Þrj0 ðjx; t0 ÞiL ;
2
ð3Þ
where r′(t∣x, t0) is the displacement at time t from its
starting position, with the mean displacement removed. The
definition of the diffusivity in equation (2) is nonlocal, since
the time lag t needs to be long enough for the diffusivity to
converge.
C06006
[15] Equation (1) requires that the float velocities be
separated into mean and eddy components [Davis, 1987].
The time‐averaged Eulerian mean velocities are highly
inhomogeneous in space. Therefore, if we remove a spatially uniform average from each float velocity, we are left
with a residual due to the mean shear that will imply dispersion and may dominate the diffusivity estimate [Bauer et
al., 1998]. As shown by Oh et al. [2000], in a shear flow the
along‐stream diffusivity is affected by the shear dispersion.
Oh et al. [2000] and Zhurbas and Oh [2003, 2004] assume
that the velocity field can be decomposed into the large‐
scale inhomogeneous mean and a slowly varying and locally
isotropic eddy field, implying that eddy mixing could be
represented by a single scalar diffusivity. In the presence of
shear flow, they suggest using the minor principal component of the diffusivity tensor ij as a robust diffusivity that
remains unbiased by the shear. This works if the mean shear
is homogeneous over the region or bin considered, which is
not the case here. Instead, we regard the diffusivity estimated from the local projection across streamlines as a
robust diffusivity that ideally is not influenced by the shear.
The residual velocity u′ from equation (1) is estimated by
subtracting the spatially varying Eulerian 1998–1999 time
mean velocities from each individual float velocity, which
minimizes the shear dispersion in the along‐stream direction. This approach works well with numerical model output
but would likely be difficult to implement using in situ float
observations for which no Eulerian mean is available. We
then follow an approach similar to that of LaCasce [2000],
who projected the individual float velocities across and
along f/H contours. We project each residual float velocity
locally across and along the spatially varying Eulerian
1998–1999 time mean velocities. By construction this
ensures that the cross‐stream dispersion by the mean is zero.
4. Diffusivities and Dispersion as a Function
of Time Lag
4.1. Long‐Time Dispersion and Diffusivities
[16] The purpose of this section is to explore the convergence properties of the Lagrangian diffusivities and
discuss their sensitivity to the choice of mean flows,
streamline projections and rotational components of the
flow. The subtraction of the Eulerian mean velocity from
each float velocity allows us to consider the long‐time dispersion of each deployment in both the along‐stream and
cross‐stream direction (section 4.1). In oceanic flows with
finite eddy scales, equation (1) and (3) entail two regimes of
dispersion. Initial dispersion is characterized by a diffusivity
that depends on the time lag. After some time TL, the dispersion is in a random walk regime, which means it is
described by a constant diffusivity ∞ and a dispersion that
is linear with the time lag. The long‐time dispersion allows
us to assess whether a diffusive limit can be reached and
when this occurs. We consider the longest possible time lag
and largest number of floats available for each patch and
deployment depth. Here we analyze Period 1 only, since
trajectories can then be considered to be on the same isopycnal for each patch and deployment depth.
[17] Figure 3 illustrates the influence of coordinate orientation on the long‐time dispersion of the Patch 3 300 m
deployment (Period 1). As LaCasce [2000] found, the dis-
5 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
C06006
Figure 3. Long‐time (1045 days) mean square displacement for Patch 3 for the 300 m deployments,
(a) in the cross‐stream (solid line) as compared to meridional (dashed line) direction and (b) in the along‐
stream directions (solid line) as compared to zonal (dashed line) direction. Also shown in Figure 3b is
the along‐stream dispersion that arises when no mean is subtracted (dotted line). Shaded are 2× standard
deviations from bootstrapping by subsampling the floats for each deployment with replacement. Note that
in the cross‐stream direction the dispersion by the mean is zero by construction.
persion is more anisotropic when it is decomposed into
cross‐stream and along‐stream components rather than
meridional and zonal coordinates (Figure 3). The long‐time
dispersion in the along‐stream direction is about an order of
magnitude larger than in the cross‐stream direction for the
Patch 3 300 m deployment. The subtraction of the local
Eulerian mean velocities does not entirely eliminate shear
dispersion, especially for the 300 m deployments. However,
the 800 and 1500 m deployments are close to a random walk
even in the along‐stream direction (Figures 5e–5h) and it
seems likely that a single scalar diffusivity will not be able
to describe the eddy mixing even if shear dispersion is
reduced. Anisotropy in particle spreading that cannot be
attributed to shear dispersion was also recently found in
particle dispersion in the North Atlantic [Kamenkovich et
al., 2009]. There is also a larger decrease in dispersion
with depth in the meridional direction than in the cross‐
stream direction (not shown). The high along‐stream values
of the dispersion leak into the meridional direction whenever the mean flow has a substantial meridional component.
The magnitude of the dispersion and diffusivities is therefore sensitive to the streamlines used in the projection.
When projected across filtered barotropic streamlines (used
for binning and along‐streamline averages in sections 4 and
5) the cross‐stream dispersion by the mean flow is still
substantial. This leads to long‐time cross‐stream eddy diffusivities about 2 times larger than when projected across
the local Eulerian mean velocity. This is because of the high
filamentation of the local time mean streamlines, and the
fact that floats do not locally follow the barotropic streamlines at all depths. The influence of the mean shear is evident in the along‐stream dispersion when no mean is
subtracted (dotted line in Figure 3b). The dispersion due to
the mean is very large and grows with time lag. It is
therefore crucial to subtract the spatially inhomogeneous
Eulerian mean velocities in order for the along‐stream
component of the dispersion to be close to linear (solid line
in Figure 3b).
[18] Topography also influences long‐time dispersion.
Whereas Patch 3 floats are constrained little by topography
and travel relatively freely across the abyssal plains west of
Drake Passage, the Patch 1 300 m cross‐stream dispersion is
relatively constant for time lags of 500–1200 days, indicative of the floats passing through the narrow gap around
Kerguelen Plateau (Figure 4a).
[19] For most deployments, the cross‐stream diffusivity
converges to a constant within the error bars after about
100 days, consistent with the random walk regime
(Figures 5a–5d). The exceptions are the Patch 1 300 m
deployments and the Patch 3 1500 m deployments, where
the diffusivity tends to zero after a time lag of 600 days.
Cross‐stream diffusivities are mostly below 1000 m2 s−1
with some variation from one deployment to another. Patch
2 exhibits the smallest cross‐stream diffusivities around
500 m2 s−1 for all three depths, indicating that most floats
are forced to go southward around Kerguelen Plateau without
much chance to disperse freely. There is not much depth
dependence detectable in the cross‐stream diffusivities.
[20] The along‐stream diffusivities (Figures 5e–5h) show
more depth dependence than the cross‐stream diffusivities
(Figures 5a–5d). This may be an indication of shear dispersion by the mean flow still playing a role, especially for
the 300 m deployments, since mean flow decreases with
depth. The along‐stream diffusion for the 800 and 1500 m
deployments can be described as a random walk from days
250 to 600, except for the Patch 2 and Patch 4 800 m
deployments. The along‐stream diffusivities for the 300 m
deployments tend to increase and decrease with time lag and
exhibit less of a random walk regime. The best convergence
properties for the along‐stream components within the first
600 days are reached again for the Patch 3 deployments for
all three deployment depths.
[21] For both cross‐stream and along‐stream components
in the Patch 1 deployments, the dispersion for the first 250–
600 days is well simulated by a constant diffusivity determined at a time lag of 100 days (Figure 4). For all patches
and time lags longer than about 600 days, cross‐stream
6 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
C06006
Figure 4. Long‐time (1045 days) mean square displacement for Patch 1 for the (a and b) 300, (c and d)
800, and (e and f) 1500 m deployment depths in the (left) cross‐stream and (right) along‐stream direction.
The mean square displacement is compared to the displacement arising when the group of floats disperses
with a constant diffusivity taken at time lag 100 days, hr′2iTaylor = 2 (100d) t (solid lines). The dashed
lines and shading are 2s errors.
dispersion tends to be overestimated by the diffusivity at
100 days time lag (see auxiliary material).1 Hence, one
constant diffusivity does not describe the dispersion for the
whole 1045 days, as will be shown in section 5.1, when
trajectories are binned geographically.
4.2. Binned Diffusivities as a Function of Time Lag
[22] To resolve the spatial variability of ij, we subdivide
the domain into bins with the expectation that the bins are
larger than the typical eddy scales thus allowing the diffusivities to converge. On the basis of the results from section
4.1, we use a maximum time lag of 90 days, just large
enough to be in the random walk regime and small enough
to resolve spatial variability.
[23] ACC streamlines are influenced by topography and
are therefore not zonally oriented in many areas. We use
bins of 10° longitude in the zonal direction, bounded by
7 geostrophic streamlines in the meridional direction.
Streamline bins use a natural coordinate system and make it
straightforward to average circumpolarly along streamlines.
The Eulerian mean velocities used for the projection of the
float velocities are spatially nonuniform and cannot be used
for constructing streamline bins. Time‐averaged barotropic
streamlines, Yg = gf−1h, computed using the model’s 1998–
1999 mean sea surface height h, also meander substantially
and may cross the same longitude multiple times. Therefore,
1
Auxiliary materials are available in the HTML. doi:10.1029/
2009JC005821.
the barotropic streamlines are filtered with a 51‐point triangular filter (equivalent to 5 degrees). In Figure 1, the
southern edge of the ACC corresponds to the −10 × 104 m2
s−1 streamline, and the northern edge corresponds to the zero
streamline. The streamline spacing is DYg = 2 × 104 m2 s−1.
This results in 6 streamline bins, denoted from south to
north as bins 1–6. In the following we refer to streamlines
1–2 as south of the Polar Front, streamlines 3–4 as around
the Polar Front or core of the ACC, and streamlines 5–6 as
north of the Subantarctic Front.
[24] Each daily float observation in each bin is treated as a
deployment point from which the float is tracked with
positive time lag (arriving at the point) and negative time lag
(leaving the point) [Davis, 1987, 1991; Poulain et al., 1996;
McClean et al., 2002] (also see Appendix A). The
Lagrangian average h iL in (1) is then the average over all
points ending up in the bin.
[25] Floats were released in the core of the ACC, so one can
expect the float density to be higher there (e.g., Figure 6a). On
the other hand, velocities are highest in the core of the ACC,
and streamline bins are therefore smaller (Figure 6b), so the
floats do not stay in the bins for long, decreasing the number
of float points in these bins. The highest number of float days
occurs around Kerguelen Plateau at 70–90°E (Figure 6a),
where velocities are small and bin sizes large.
[26] Diffusivites are calculated from equation (1) for
positive and negative time lags for all bins. Examples of the
cross‐stream (Figure 7) and along‐stream (Figure 8) diffusivities as a function of time lag are shown for Patch 4 floats
7 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
Figure 5. (a–d) Long‐time (1045 days) cross‐stream diffusivities for all Patches at 300 (black), 800 (red),
and 1500 m (blue) deployment depths and (e–h) the same for the along‐stream direction. Shaded regions
are 2s errors from bootstrapping by subsampling the floats for each deployment with replacement.
Figure 6. (a) Number of float points per bin for 800 m deployments. (b) Average Eulerian mean velocity
(cm s−1) in each bin for the 800 m deployments and trajectories after 1045 days (Period 1). The average
was obtained by first interpolating the 2 year Eulerian mean velocity to each float point and then averaging over the bin. Bins are defined at the beginning of section 4.2.
8 of 18
C06006
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
C06006
Figure 7. Cross‐stream diffusivities as a function of time lag for the Patch 4 at 300 (black), 800 (red),
and 1500 m (blue) deployments for streamline bin 3 and longitude bins (a) 290–300°E, (b) 300°–310°E,
(c) 310°–320°E, (d) 320°–330°E, (e) 330°–340°E, and (f) 340°–350°E. Also shown are 2s uncertainties
from the bootstrapping. Vertical gray lines indicate the diffusivity at time lag ±75 days.
that are in streamline bin 3 around Drake Passage.
Uncertainties are estimated using a bootstrapping process.
For each patch and deployment depth, the 160 floats are
subsampled 100 times [Efron and Gong, 1977]. The subsample has 160 floats, allowing for duplicates. The standard
deviations shown in Figures 7 and 8 represent 95% significance levels derived from the subsampling. Generally, the
diffusivities in these bins reach a constant value by time lag
±75 days. However, for some bins, the along‐stream diffusivites from the 300 m deployments continue increasing
after 75 days (Figures 8a–8c). Cross‐stream diffusivities
show overshoots within the first 10 days and some oscillating behavior particularly for the 300 m deployments
(Figure 7) indicating negative lobes in the autocovariance
functions (discussed below).
4.3. Defining k∞
[27] Lagrangian diffusivity represents diffusion accurately
in the ideal scenario of finite scale eddies and homogeneous
statistics. In the realistic scenario considered here, both
spatial inhomogeneity and nonstationarity introduce potential biases in the estimates of the horizontal and vertical
eddy diffusivity distributions.
[28] As discussed in sections 4.1 and 4.2, the along‐
stream velocity autocovariance integral converges only if we
subtract the spatially varying time mean instead of a constant bin mean. Moreover, the eddy field contains rotational
components [e.g., Marshall and Shutts, 1981], leading to
looping and meandering trajectories, that influence the tail
of the velocity autocovariances [see, e.g., Berloff and
McWilliams, 2002a, 2002b; Veneziani et al., 2004, 2005a,
2005b]. As discussed by a number of authors [e.g., Marshall
and Shutts, 1981; Jayne and Marotzke, 2002; Eden et al.,
2007; Griesel et al., 2009], the rotational parts of the eddy
fluxes do not contribute to the local heat budget. Therefore,
they should have no net particle dispersion and zero
Lagrangian diffusivity ∞.
[29] When floats travel through meanders or circle around
stationary eddies, this leads to oscillatory patterns in the
autocovariance and diffusivity, as illustrated in Figure 9.
This effect was also discussed by Berloff and McWilliams
[2002a] and by Veneziani et al. [2004, 2005a, 2005b].
Figures 9a–9c considers an idealized scenario of particles
circling twice around eddies and meandering in a half circle.
These rotational motions do not contribute to the diffusivity
at time lags longer than the meander or circling time scale.
Superimposed on this rotational field is a random velocity
field with a decorrelation scale of 10 days. For the half circle
meanders, diffusivity as a function of time lag grows first,
reaches a maximum and then declines again before it returns
(by time lag 10 days) to the correct asymptotic value as
determined only by the random component of the velocity
field (Figure 9c). The net cross‐stream dispersion of the
float due to the meanders and circles is zero. In a more
9 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
C06006
Figure 8. Along‐stream diffusivities as a function of time lag for the Patch 4 at 300 (black), 800 (red),
and 1500 m (blue) deployments for streamline bin 3 and longitude bins (a) 290–300°E, (b) 300°–310°E,
(c) 310°–320°E, (d) 320°–330°E, (e) 330°–340°E, and (f) 340°–350°E. Also shown are 2s uncertainties
from the bootstrapping. Vertical gray lines indicate the diffusivity at time lag ±75 days.
realistic scenario of a complex eddy field that consists of
eddies of different sizes that grow and decay (Figure 9d),
velocity autocovariance functions will show more complex
oscillatory behavior (Figure 9e). One question is whether the
oscillatory behavior is transient and the diffusive limit can
be reached, as in the idealized scenario of Figures 9a–9c and
in the long‐time dispersion. If more independent trajectories
are considered in the Lagrangian average, then the oscillations in the tail of the autocovariance are more likely to
cancel each other. The full group of floats in section 4.1 for
example, shows little oscillatory behavior after about
50 days (Figure 10). However, there still is a maximum
diffusivity in the 0–10 day time lag interval that is about
2–4 times larger than ∞ (Figure 10). This maximum may
reflect the scale of the meandering current and does not
disappear or average out when more and more floats are
considered that are subject to the same meander scale.
[30] A standard practice to determine ∞ in equation (2) is
to integrate the autocovariance functions to the first zero
crossing [e.g., Freeland et al., 1975; Krauss and Böning,
1987; Poulain and Niiler, 1989; Lumpkin et al., 2001].
This works if the trajectories in the eddy field are not
dominated by looping or meandering patterns. Alternatively,
idealized covariance functions that asymptote to zero are
fitted to the observed covariance functions [e.g., Garaffo et
al., 2001; Sallée et al., 2008]. Lumpkin et al. [2001] com-
pared several different methods to determine ∞. They
found that the integration of the autocovariance until the
first zero crossing leads to higher eddy scale estimates in the
region of the Gulf Stream than when an idealized autocovariance consisting of exponentially decaying and wave
components is fitted to the data and integrated to infinity.
The integration of RL until the first zero crossing is equivalent to taking the maximum of the diffusivity as a function
of time lag in the cross‐stream direction. As shown in
Figure 9f, the maximum of the diffusivity is not the right
measure for the dispersion. The negative and positive lobes
after the maximum cannot be neglected in the integration
of the covariance.
[31] The most accurate determinations of the asymptotic
diffusivity are made by estimating (t) at the smallest lag
for which (t) = ∞ [e.g., Davis, 1991; Swenson and Niiler,
1996]. We therefore estimate the diffusivity ∞ for all bins
as an average of the diffusivity at time lags +75 days and
−75 days. The 75 day integral is chosen as it represents the
smallest timescale when most of the diffusivities for the bins
are expected to be in the random walk regime and to
have reached a plateau (e.g., Figure 10, and compare also
Figure 5). This timescale is substantially larger than the
≈5–10 day timescales typical of the first zero crossing of
our Lagrangian cross‐stream velocity autocovariance.
10 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
C06006
Figure 9. (left) Idealized scenario; random velocity field plus constant mean flow in along‐stream direction, superimposed on particles passing through a half circle meander (black) and circling twice around a
stationary eddy (red) (see Appendix B for details). (a) Idealized trajectories. (b) Velocity autocovariance
in cross‐stream direction for particles passing through the half circle meander (black), circling around the
eddy twice (red) and the average of these two scenarios (blue). (c) Corresponding cross‐stream diffusivities as a function of time lag. (right) Examples from the POP float deployments. (d) Examples of
circling and meandering numerical trajectories from the Patch 2 at 300 m deployment. (e) Corresponding
velocity autocovariance. (f) Diffusivities as a function of time lag.
[32] Unlike the cross‐stream diffusivities (Figure 7), the
along‐stream diffusivities (Figure 8) do not exhibit oscillatory behavior, and the maximum in the 0–90 day time lag
interval is equal to ∞
k . This may be due to the fact that while
a full circle trajectory does not lead to any net dispersion in
the along‐stream direction, meandering trajectories do lead
to net along‐stream dispersion and no oscillatory behavior.
On average, meanders, rather than the circling trajectories,
may dominate the time lag dependence of the Lagrangian
diffusivities. Cross‐stream diffusivities are much smaller than
along‐stream diffusivities and the strong rotational effects are
more likely to overwhelm the cross‐stream diffusivities.
5. Lagrangian Diffusivities and Their
Relationship to Eddy Kinetic Energy, Eddy Scales,
and Mean Flow
[33] While the results of section 4 show that cross‐stream
diffusivities converge to stable values over 100–1000 day
average intervals, averaging along extended trajectories
does not allow us to consider spatial patterns or to examine
how they may vary as a function of depth. In order to assess
geographic variation in diffusivity, in section 5.1 we bin
data geographically, and in order to assess vertical struc-
Figure 10. Enlargement for the first 100 days of the cross‐
stream diffusivities as a function of time lag for Patch 4
(Figure 5d).
11 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
Figure 11. (left) Cross‐stream and (right) along‐stream diffusivities (m2 s−1) for the deployment depths
of (a and d) 300 m (0 = 27.36 ± 0.61 kg m−3), (b and e) 800 m (0 = 27.58 ± 0.27 kg m−3), and (c and f)
1500 m (0 = 27.74 ± 0.20 kg m−3) for 1 to 1045 day trajectories (Period 1). In a few cases where bins of
different deployments overlapped, diffusivities were averaged. Standard errors for the cross‐stream diffusivities are on average 30% of the diffusivity for the 300 m deployments and about 10% of the diffusivity
for the 800 and 1500 m deployments. Standard errors for the along‐stream diffusivities are on average
20% of the diffusivity for the 300 m deployments and about 10% of the diffusivity for the 800 and
1500 m deployments.
12 of 18
C06006
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
Figure 12. Correlations between
pffiffiffiffiffiffiffiffiffi cross‐stream diffusivities
() and eddy velocities ( hu0 2 i) as a function of mean
depth, under consideration of all trajectories (Period 1
and 2). R2 is the variance explained by the linear regression
model devided by the total variance of the data. Solid line is
∞. Dotted line is max. Here max is obtained by taking the
maximum of the diffusivity in the 0–75 day time lag interval, which in the cross‐stream direction is equivalent to integrating the Lagrangian autocovariance RL to the first zero
crossing. Error bars are standard errors from bootstrapping
by subsampling the floats in each depth interval with
replacement.
ture in section 5.2 we sort the float trajectories into 4 depth
intervals.
5.1. Horizontal Distributions
[34] By averaging the data in geographic bins, we are able
to evaluate horizontal distributions of the diffusivities ∞ at
each deployment depth. Since there is a discontinuity in the
density field at day 1045, we consider only the trajectories
for Period 1. This means that horizontal distributions of
diffusivities are on isopycnal surfaces that are relatively
constant for each of the deployments (compare Figure 2).
Only bins with a minimum of 1500 daily float positions
are considered, ensuring ‘convergence’ properties as in
Figures 7 and 8. When fewer than 1500 float positions are
available we find that convergence is not reliably achieved,
especially for the 300 m deployments. Since most floats do
not cross the Subantarctic Front, streamline bin 6 only has
significant data for longitudes between 180°E and 250°E
(Figure 6). After 1045 days, the floats have not traveled far
enough from any of the initial deployment patches to
sample the region south of Australia and New Zealand
(Figures 1b and 1c). Also, in some bins float density is
reduced due to mixed layer encounters in the 300 m deployments, for example 180°E–220°E (Figures 11a and 11b).
[35] Horizontal patterns of cross‐stream and along‐stream
diffusivities for the three deployment depths are inhomogeneous: the latitude dependence varies with geographic
location, and diffusivities can have local maxima and minima in the core of the ACC jet (Figure 11). Note that similar
horizontal distributions for three depth intervals, instead of
deployment depths, are found in Figure S4 in the auxiliary
material. The relatively high spatial variability in Figure 11
was not evident in the long‐time diffusivity estimates from
C06006
section 4.1. In the east Pacific sector of the Southern Ocean
where the flow crosses the abyssal plains, cross‐stream
diffusivities are high in the core of the ACC jet (see also
Figure 6). West of 0°E cross‐stream diffusivities are lower
in the core of the ACC (see also Figure S4 in the auxiliary
material). The highest cross‐stream diffusivities, exceeding
2500 m2 s−1, are found south of the Agulhas Retroflection
and near the Brazil‐Malvinas confluence zone. High values
are also found where the flow veers around islands or
topographic features, such as the Kerguelen Plateau (70°–
90°E) and Chatham Rise south‐east of New Zealand
(180°–190°E).
[36] Along‐stream diffusivities are about 3–10 times
larger than cross‐stream diffusivities (Figure 11). Along‐
stream and cross‐stream diffusivities are generally not correlated. However, areas of high along‐stream diffusivities
generally would appear to coincide with areas of enhanced
cross‐stream diffusivities if the float velocities were
represented in zonal and meridional coordinates or if they
were projected into along‐stream and cross‐stream
coordinates using filtered barotropic streamlines rather than
local time‐averaged velocities. In these cases there is leakage of the high along‐stream (or zonal) values into the
cross‐stream (or meridional) terms. The largest values of
along‐stream diffusivity, exceeding 10,000 m2 s−1, occur
in the Brazil‐Malvinas confluence zone and north of the
Kerguelen Plateau.
[37] What factors determine the horizontal distributions of
the diffusivities? Both eddy kinetic energy and eddy length
scales have been hypothesized to govern diffusivity variations. These two quantities are not independent, since eddy
length scales can depend on eddy kinetic energy. The
Lagrangian diffusivity can be thought of asp
the
product of
ffiffiffiffiffiffiffiffiffi
the Lagrangian average of the eddy velocity hu0 2 i, and the
Lagrangian eddy length scale LL, or alternatively, as the
product of the Lagrangian average of the squared velocity
i and
hu′2p
ffiffiffiffiffiffiffiffiffithe characteristic Lagrangian eddy time scale TL =
LL/ hu0 2 i. The Lagrangian eddy scales are obtained from
the integral of the velocity autocovariance. In other words,
the diffusivity is determined by the Lagrangian averaged
velocity autocovariance at zero lag and by the structure of
the tail of the Lagrangian autocovariance for time lags > 0.
If either the Lagrangian time scales or the Lagrangian length
scales could be considered constant, then the spatial distributions of the diffusivities would be determined by the
eddy velocities or eddy kinetic energies. No consistent
relationship between Lagrangian diffusivities and eddy
kinetic energy or eddy velocity has been found in numerous
studies using floats from many areas in the ocean (see, e.g.,
Lumpkin et al. [2001] for an overview), suggesting no
universal constant LL or constant TL rules.
[38] Figure 12 shows that the correlation of the
Lagrangian diffusivities with the eddy velocities can be
quite different depending on what time lag is used for the
integration of the velocity autocovariance. Only about 50%
of the horizontal distributions of ∞ can be explained by
variation of the eddy velocity in the surface depth interval,
and only about 10% for the 1500–2500 m depth interval.
Clearly, the Lagrangian eddy length scale cannot be considered constant especially for the deeper depth levels. On
the other hand, when we use max as the measure for the
13 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
C06006
Figure 13. (a) Cross‐stream diffusivities (m2 s−1), (b) along‐stream diffusivities (m2 s−1), (c) across‐
stream eddy velocity hu2?i (m s−1), (d) along‐stream eddy velocity hu2?i (m s−1), (e) cross‐stream eddy
length scale (km), and (f) along‐stream eddy length scale (km) as a function of mean bin depth. Pluses
indicate results from the 1–1045 day trajectories (Figure 1 (left)) and circles indicate bins for the 1046–
1761 days trajectories (Figure 1 (right)). The three vertical lines are along‐streamline averages as a
function of mean depth intervals 5–500, 500–1000, 1000–1500, and 1500–2500 m over the streamline
intervals 1–2 (blue, south of Polar Front), 3–4 (green, around Polar Front), and 5–6 (red, north of
Subantarctic Front) from south to north.
Lagrangian diffusivity, we find misleading high correlations
with eddy velocity, which could lead to false conclusions
that eddy length scales may be considered constant and that
eddy velocities or eddy kinetic energies may be used to
determine the spatial distributions of the diffusivities. Note
that similar correlation patterns and differences for ∞ and
max are found when the cross‐stream diffusivity is correlated with the eddy kinetic energy, instead of the eddy
velocity. Also, the along‐stream diffusivity is highly correlated with the eddy velocities or eddy kinetic energies (not
shown).
[39] Neither the eddy length nor the eddy time scale can
be considered horizontally homogeneous in the cross‐stream
direction, suggesting that parameterizations of eddy diffusivity that depend solely on eddy kinetic energy are likely to
be incomplete.
5.2. Depth Dependence
[40] We now assess the vertical structure of the diffusivities as a function of mean bin depth, without regard for the
isopycnal on which the floats were initially launched. All
float trajectories from the left and right panels of Figure 1
(Periods 1 and 2) are considered in this section. The
spread of the individual cross‐stream (Period 1 pluses and
Period 2 circles in Figure 13a) and along‐stream (Figure 13b)
diffusivity estimates as a function of depth shows that the
flow regimes vary with longitude and latitude (Figure 11). In
general, the along‐stream averaged diffusivities increase
northward, although this increase becomes less significant
with depth. As shown in the previous section, this latitude
dependence is not constant for all longitudes. At some
longitudes, local diffusivity maxima occur in the core of the
ACC jet with smaller diffusivities to the north, for example
west of Drake Passage (240°E–280°E). For the northernmost streamlines (red line in Figure 13), cross‐stream diffusivities are 1800 ± 1200 m2 s−1 for the 500–1500 m depth
interval, but in this depth range only 4 bins have significant
data (Figure 11). Since the floats travel deeper toward the
north, the 0–500 m depth interval was not sampled by the
two northernmost streamline bins. Around the Polar Front
(green lines in Figure 13), cross‐stream diffusivities of
between 500 and 1000 m2 s−1 are fairly uniform with depth.
The depth dependence is more pronounced for along‐stream
diffusivities and is what one might expect if diffusivity were
correlated with eddy kinetic energy and eddy transports,
decreasing with depth. Lagrangian diffusivities (Figures 13a
and 13b) show no evidence of a maximum at depth, in
contrast with predictions of critical layer theory [Green,
1970; Smith and Marshall, 2009].
14 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
Figure 14. Eulerian mean velocity as a function of eddy
kinetic energy averaged over each bin. Across‐streamline
diffusivity values are in color for three intervals (<500 m2
s −1 (blue), between 500 and 1000 m 2 s −1 (black), and
>1000 m2 s−1 (red)). In the upper ocean within the ACC,
um − c ≈ um. For the 1500 m deployments (triangles), using
um − c would be more appropriate.
[41] Whereas in the previous section we considered the
importance of the horizontal distributions of the eddy
velocity and eddy scales for determining the horizontal
distributions of diffusivities, we now investigate the relative
importance of the length scale and eddy velocity in determining the depth dependence of the diffusivities. A decrease
of diffusivities with depth was expected when diffusivities
were correlated with eddy kinetic energy. Figures 13c and
13e shows the cross‐stream eddy kinetic energy component
and eddy length scale and Figures 13d and 13f the along‐
stream eddy velocity and length scale as a function of mean
bin depth. The cross‐stream and along‐stream eddy kinetic
energy components (Figures 13c and 13d) are comparable in
their magnitudes and distributions, indicating isotropic
eddies. For the cross‐stream direction, the interplay between
eddy length scales increasing with depth (Figure 13e) and
eddy velocities decreasing with depth (Figure 13f) leads to
eddy diffusivities (Figure 13a) that are relatively constant
with depth. The along‐stream eddy length scales decrease
with depth, especially north of the Subantarctic Front (red
line). Overall, the analysis demonstrates that a parameterization of diffusivity that is only related to eddy kinetic
energy is incomplete and leads to the wrong depth dependence. For example, the cross‐stream eddy velocity in the
1500–2500 m depth interval is only about 30% of its near
surface value. In contrast the cross‐stream diffusivity is
about the same as its near surface value in the 1500–2500 m
depth interval around the Polar Frontal Zone.
[42] If the eddy kinetic energy alone does not determine
the cross‐stream mixing, then perhaps it is the strength of
the mean flow that limits mixing and leads to smaller eddy
scales?
[43] Shuckburgh et al. [2009a, 2009b] reported on average a significant correlation between effective eddy diffusivities and EKE/(Um − c)2 where Um is the strength of the
C06006
mean flow and c is the eddy velocity, illustrating that strong
mean flow may limit mixing, at least as long as it is not
comparable to the eddy velocity. Similarly to Figure 11 of
Shuckburgh et al. [2009b], Figure 14 shows the dependence
of the Lagrangian cross‐stream diffusivity on both the
strength of the bin‐averaged mean flow and EKE. There is
evidence that a stronger mean flow acts as a mixing barrier
within the ACC, since high diffusivities tend to be located
below the black line in Um/EKE space, and small eddy
diffusivities above the black line towards higher values of
mean flow. There are some bins for example with small
EKE (around 10 cm s−1) but large Lagrangian diffusivities.
However, cross stream dispersion is not consistently
reduced in areas with strong Eulerian mean velocities in the
ACC. As Shuckburgh et al. [2009a] found, we find that the
correlation of the cross‐stream Lagrangian diffusivities with
EKE/U2m is significant (R2 = 0.47) when all bins are taken
into account, and slightly lower when the cross‐stream
diffusivity is just correlated with EKE (cf. Figure 12). On
the other hand, the correlation of max with EKE/U2m is only
0.32, whereas the correlation with EKE alone is high at 0.76
when all bins are taken into account (cf. Figure 12). Hence,
max does not detect mixing barriers where the mean flow is
strong.
[44] We note that the floats stay mainly within the
boundaries of the ACC, and, as mentioned, north of the
Subantarctic Front there are no bins in the 5–500 m depth
interval and only 4 bins in the 500–1500 m depth interval
with statistically significant diffusivities. Therefore, we do
not adequately resolve the area north of the ACC, where the
mean flow is much weaker and likely Lagrangian diffusivities will be larger in the along‐streamline average.
6. Conclusions
[45] We have used numerical floats to estimate horizontal
and vertical distributions of Lagrangian diffusivities in the
Southern Ocean. Lagrangian cross‐stream diffusivities
converged to a constant value, provided that the time lag
was long enough and that oscillations were minimized by
considering enough float points in the Lagrangian average.
Along‐stream diffusivities also tended to converge, at least
for the 800 and 1500 m deployments, provided that a spatially varying time mean was subtracted from the float
velocities to minimize shear dispersion. Our results have
shown that diffusivities are anisotropic and that a single
scalar diffusivity is not appropriate to describe eddy mixing
in the ACC. The influence of topography was clearly evident in the long‐time dispersion, where a single constant
diffusivity did not simulate the dispersion for the whole
Period 1, except in the Patch 3 300 and 800 m deployments
that were the least constrained by topography.
[46] The diffusivity distributions were analyzed in the
context of theories predicting specific depth and latitude
dependence. Lagrangian diffusivities are horizontally inhomogeneous and can be high in regions where eddy kinetic
energy is high, such as in the Polar Front and to the north of
the Subantarctic Front. But the horizontal distributions of
diffusivities are not highly correlated with the eddy kinetic
energy or eddy velocity, indicating that the Lagrangian eddy
length and time scales cannot be considered constant. We
have pointed out the substantial differences in cross‐stream
15 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
Lagrangian diffusivities between at the first zero crossing
and ∞. Not only do the magnitudes of the diffusivities
differ between these two estimates but also their spatial
distributions. There is a decrease with depth when the
maximum is taken, because the diffusivities, as a function
of time lag, tend to show more oscillatory behavior close to
the surface than at depth. In addition, high correlations with
eddy kinetic energy in previous studies [Krauss and Böning,
1987; Lumpkin et al., 2001; Zhurbas and Oh, 2004] might
occur because these studies used the maximum of the diffusivities rather than ∞. With the more appropriate measure
∞, we do not find such high correlations with eddy kinetic
energy. Parameterizations that only use eddy kinetic energy
to parameterize the diffusivities are therefore incomplete.
The Lagrangian near surface cross‐stream diffusivities in the
core of the ACC jet are more comparable to the Marshall et
al. [2006] and Abernathy et al. [2009] effective diffusivity
estimates of close to 500 m2 s−1. Our diffusivity in the core
of the ACC is sensitive to the way the float velocities are
projected. We required that the cross‐stream dispersion by
the mean be zero.
[47] Our cross‐stream diffusivities are substantially
smaller than those found by Sallée et al. [2008] within the
ACC. This might be due to the fact that Sallée et al. [2008]
considered mixing at the surface which includes stronger
diapycnal mixing that occurs within the mixed layers and
has been excluded in our analysis. Further, they used a
different method to separate along‐stream and cross‐stream
mixing. They determined the orientation of the cross‐stream
direction by using the minor axis of the velocity covariance
tensor in each bin. We have taken into account small‐scale
spatial variation of the Eulerian velocity by projecting across
and along the local mean velocity at each point, which
greatly reduces the cross‐stream diffusivity.
[48] When the time lag is long enough, our Lagrangian
diffusivities are correlated with the eddy kinetic energy to
mean flow ratio, albeit with a somewhat lower correlation
coefficient than for the effective diffusivities of Shuckburgh
et al. [2009a, 2009b]. The correlation with EKE alone is
weaker. Our mean flow represents averages over 10° longitude bins, and there are gaps in the horizontal distribution
of the diffusivities due to insufficient Lagrangian data.
Therefore, a clearer distinction between stronger and weaker
mean flows, which lead to smaller and larger cross‐stream
dispersion respectively, may not be detected with the single
particle dispersion considered here.
[49] Cross‐stream eddy diffusivities decrease with depth
north of the Subantarctic Front, although the high values
close to the surface are only found in a few bins. Along‐
stream averages of cross‐stream eddy diffusivities around
the Polar Front are relatively uniform with depth. We did
not find support for enhanced mixing due to critical layers
below the core of the ACC jet. However, cross‐stream eddy
length scales did increase with depth, masked by the strong
depth decrease of eddy velocities. This might just indicate
that eddies are larger at depth but might not mean there is
more mixing. Naveira Garabato et al. [2009] have recently
argued that the eddy mixing length, as defined by, for
example, Armi and Stommel [1983], determines the spatial
distributions of the eddy diffusivities. The question remains
how classical eddy mixing lengths and Lagrangian eddy
length scales are related.
C06006
[50] The depth dependence of our diffusivities is different
from the depth dependence of the effective diffusivities
obtained from Abernathy et al. [2009], who found enhanced
values up to 4000 m2 s−1 below the core of the ACC jet, at
depths of 1500 m on average, and small values of about
500 m2 s−1 in the core of the jet. While Lagrangian and
effective diffusivities have similar magnitudes and distributions in the upper ocean, it is unclear why they differ at
depth. The effective diffusivity may capture small‐scale
features due to topographic interactions that the Lagrangian
diffusivity can not resolve due to its nonlocalness. If the
eddy mixing at depth is highly localized, we simply may not
sample the depth interval 1500–2500 m well enough due to
the limited amount and extent of the numerical trajectories.
Some of the high effective diffusivities from Abernathy et
al. [2009] occur at depths much deeper than 2500 m, a
region that we do not sample at all. Furthermore, in the
effective diffusivity method, the passive tracer follows the
instantaneous streamlines, and instantaneous cross‐stream
mixing is evaluated where the rotational part of the eddies
plays no role. The tail of the velocity autocovariance for
time lags > 0, and therefore the calculation of the
Lagrangian eddy scales, is biased by the presence of rotational components and by the problem of identifying
appropriate eddy length scales. Griesel et al. [2009] have
recently evaluated the curl and divergence of the horizontal
eddy heat flux in 1/10° POP and found that the rotational
components, as measured by the curl, dominate even for
averages over scales much larger than the size of the eddies.
Extended numerical float deployments are needed in the
future to sample all latitudes and depths in the Southern
Ocean, and to evaluate further whether the bias of the
rotational part can indeed be eliminated in the Lagrangian
diffusivity.
Appendix A
[ 51 ] The elaborated flux versus gradient law of Davis
[1987, 1991] and Poulain et al. [1996]
hu0i ðx; t Þ0 ðx; tÞi ¼ Z
ij ðx; Þ ¼
0
Z
t
0
d@ ij ðx; Þ@xj ðx; t Þ
d ~hu0i ðt0 jx; t0 Þ u0j ðt0 þ ~jx; t0 ÞiL
ðA1Þ
ðA2Þ
As !1 :
ðA3Þ
hu0i ðx; t Þ0 ðx; tÞi ¼ 1
ij ðxÞ@xj ðx; t Þ
ðA4Þ
specifies the flux of a (passive) tracer Q from the recent
history of its mean concentration. The driving force for the
eddy flux at time t is the sum of previous Q gradients at
times t − t, weighted by the time derivative of at time
range t. Here is defined as the integral of the time lagged
Lagrangian velocity autocovariance. This is an elaboration
of the familiar advection‐diffusion equation, where the
elaboration reflects the finite scales of the eddies. If t goes
to infinity, or, rather, is greater than some time T reflecting
the finite scales of the dispersing eddies, (x, t) approaches a
16 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
C06006
Figure A1. (left) Ideal scenario of an ensemble of particles passing through x at time t0, which becomes
in practice (right) the scenario of floats passing though a space domain or bin. Each daily float observation
in the domain is considered as a deployment point from which the float is tracked with positive time lag
(arriving at the point) and negative time lag (leaving the point).
constant value ∞ and the eddy flux is approximated by the
typical advection‐diffusion equation. The Lagrangian average is an average over all particles passing through x, to. In
theory, we are looking at an ensemble of particles passing
through x at time t0 (Figure A1 (left)). In practice with only
one realization of the simulated floats, we are looking at
floats passing through a space domain or bin (Figure A1
(right)). Each daily float observation in the domain is considered as a deployment point from which the float is
tracked with positive time lag (arriving at the point) and
negative time lag (leaving that point). Lagrangian average is
then an average over all points passing through the bin.
Appendix B
[52] For illustration purposes we have carried out tests to
show the impact of circular eddies on the Lagrangian diffusivity estimates. More detailed analyses with looping
trajectories have been carried out by, e.g., Veneziani et al.
[2004, 2005a, 2005b]. We considered the idealized situation of particles in a random velocity field circling around an
idealized eddy. The cross‐stream velocity as a function of
time is modeled as the sum of a cosine function and a
random component
u? ðntÞ ¼ rðnt Þ cosðnt Þ þ urand ðnt Þ
ðB1Þ
ujj ðnt Þ ¼ rðnt Þ sinðnt Þ þ urand ðnt Þ þ um
ðB2Þ
where urand (nDt) = fm?x (nDt). Here, um is a constant mean
flow that does not play any role in the cross‐stream dispersion. Random numbers x are generated to have a normal
distribution with standard deviation s, and then filtered with
a boxcar filter fm with m points. The variance s2 determines
the value to which the diffusivity will converge. The filter
scale m determines the time lag when the diffusivity has
converged. For the particles to loop around a circle the
radius r is set to r0 for some duration tc to tc + 4p to circle
twice, and tc to tc + p for the half circle meander, and is zero
otherwise.
[ 53 ] Acknowledgments. This research was supported by the
National Science Foundation grants OCE‐0549225, OCE‐9985203/OCE‐
0049066, and OPP‐0337998 by the National Aeronautics and Space
Administration contract EOS/03‐0602‐0117 and the Office of Science
(BER), U.S. Department of Energy, grant DE‐FG02‐05ER64119. The
global simulation was carried out as part of a Department of Defense High
Performance Computing Modernization Program (HPCMP) grand challenge grant at the Maui High Performance Computing Center (MHPCC).
A.G. thanks Stefan Riha for useful discussions and comments.
References
Abernathy, R., J. Marshall, E. Shuckburgh, and M. Mazloff (2009), Studies
of enhanced isopycnal mixing at deep critical levels in the Southern
Ocean, J. Phys. Oceanogr., 40, 170–184.
Armi, L., and H. Stommel (1983), Four views of a portion of the North
Atlantic subtropical gyre, J. Phys. Oceanogr., 13, 828–857.
Bauer, S., M. S. Swenson, A. Griffa, A. J. Mariano, and K. Owens (1998),
Eddy‐mean flow decomposition and eddy‐diffusivity estimates in the
tropical Pacific Ocean: 1. Methodology, J. Geophys. Res., 103(C13),
30,855–30,871.
Berloff, P. S., and J. C. McWilliams (2002a), Material transport in oceanic
gyres. Part I: Phenomenology, J. Phys. Oceanogr., 32, 764–796.
Berloff, P. S., and J. C. McWilliams (2002b), Material transport in oceanic
gyres. Part II: Hierarchy of stochastic models, J. Phys. Oceanogr., 32,
797–830.
Bower, A. S., H. T. Rossby, and J. L. Lillibridge (1985), The Gulf Stream—
Barrier or blender?, J. Phys. Oceanogr., 15, 24–32.
Bryden, H. L., and S. A. Cunningham (2003), How wind‐forcing and air‐
sea heat exchange determine the meridional temperature gradient and
stratification for the Antarctic Circumpolar Current, J. Geophys. Res.,
108(C8), 3275, doi:10.1029/2001JC001296.
Cerovecki, I., R. A. Plumb, and W. Heres (2009), Eddy transport and mixing in a wind and buoyancy driven jet on a sphere, J. Phys. Oceanogr.,
39, 1133–1149.
Danabasoglu, G., and J. Marshall (2007), Effects of vertical variations of
thickness diffusivity in an ocean general circulation model, Ocean Modell.,
18, 122–141.
Davis, R. (1987), Modelling eddy transport of passive tracers, J. Mar. Res.,
45, 635–666.
Davis, R. (1991), Observing the general circulation with floats, Deep Sea
Res., 38, suppl. 1, S531–S571.
Davis, R. (2005), Intermediate‐depth circulation of the Iindian and South
Pacific oceans measured by autonomous floats, J. Phys. Oceanogr., 35,
683–707.
Döös, K., and D. Webb (1994), The Deacon cell and other meridional cells
of the Southern Ocean, J. Phys. Oceanogr., 24, 429–442.
Eden, C. (2006), Thickness diffusivity in the Southern Ocean, Geophys.
Res. Lett., 33, L11606, doi:10.1029/2006GL026157.
Eden, C., R. J. Greatbatch, and D. Olbers (2007), Interpreting eddy fluxes,
J. Phys. Oceanogr., 37(5), 1282–1296.
Efron, B., and G. Gong (1977), A leisurely look at the bootstrap, the jackknife, and cross‐validation, Am. Stat., 37, 36–48.
Ferreira, D., J. Marshall, and P. Heimbach (2005), Estimating eddy stresses
by fitting dynamics to observations using a residual‐mean ocean circulation model and its adjoint, J. Phys. Oceanogr., 35, 1891–1910.
Freeland, H., P. Rhines, and T. Rossby (1975), Statistical observations of
the trajectories of neutrally buoyant floats in the North Atlantic,
J. Mar. Res., 33, 383–404.
Garaffo, Z. D., A. J. Mariano, A. Griffa, C. Veneziani, and E. P. Chassignet
(2001), Lagrangian data in a high‐resolution numerical simulation of the
North Atlantic, J. Mar. Syst., 29, 157–176.
Gent, P. R., and J. C. McWilliams (1990), Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., 20, 150–155.
Green, J. S. (1970), Transfer properties of the large‐scale eddies and the
general circulation of the atmosphere, Q. J. R. Meteorol. Soc., 96,
157–185.
Griesel, A., S. T. Gille, J. Sprintall, J. L. McClean, and M. E. Maltrud
(2009), Assessing eddy heat flux and its parameterization: A wavenumber perspective from a 1/10° ocean simulation, Ocean Modell., 29, 248–
260, doi:10.1016/j.ocemod.2009.05.004.
Griffies, S. M. (1998), The Gent‐McWilliams skew flux, J. Phys. Oceanogr., 28, 831–841.
17 of 18
C06006
GRIESEL ET AL.: LAGRANGIAN DIFFUSIVITIES IN ACC
Jayne, S. R., and J. Marotzke (2002), The oceanic eddy heat transport,
J. Phys. Oceanogr., 32, 3328–3345.
Johnson, G. C., and H. L. Bryden (1989), On the size of the Antarctic
Circumpolar Current, Deep Sea Res., 36, 39–53.
Kamenkovich, I., P. Berloff, and J. Pedlosky (2009), Anisotropic material
transport by eddies and eddy‐driven currents in a model of the North
Atlantic, J. Phys. Oceanogr., 39, 3162–3175.
Keffer, T., and G. Holloway (1988), Estimating Southern Ocean eddy flux
of heat and salt from satellite altimetry, Nature, 332, 624–626.
Killworth, P. D. (1997), On the parameterization of eddy transfer. Part I:
Theory, J. Mar. Res., 55, 1171–1197.
Krauss, W., and C. Böning (1987), Lagrangian properties of eddy fields in
the northern North Atlantic as deduced from satellite‐tracked buoys,
J. Mar. Res., 45, 259–291.
LaCasce, J. H. (2000), Floats and f/H, J. Mar. Res, 58, 61–65.
LaCasce, J. H. (2008), Lagrangian statistics from oceanic and atmospheric
observations, in Transport and Mixing in Geophysical Flows, edited by
J. B. Weiss and A. Provenzale, pp. 165–218, Springer, Berlin.
Large, W. G., J. C. McWilliams, and S. C. Doney (1994), Oceanic vertical
mixing: A review and a model with a nonlocal boundary layer parameterization, Rev. Geophys., 32(4), 363–403.
Lumpkin, R., A.‐M. Treguier, and K. Speer (2001), Lagrangian eddy scales
in the Northern Atlantic Ocean, J. Phys. Oceanogr., 32, 2425–2440.
Maltrud, M. E., and J. L. McClean (2005), An eddy resolving global 1/10°
ocean simulation, Ocean Modell., 8 (1–2), 31–54.
Marshall, J., and T. Radko (2003), Residual mean solutions for the Antarctic
Circumpolar Current and its associated overturning circulation, J. Phys.
Oceanogr., 33(11), 2341–2354.
Marshall, J., and G. Shutts (1981), A note on rotational and divergent eddy
fluxes, J. Phys. Oceanogr., 11, 1677–1680.
Marshall, J., D. Olbers, H. Ross, and D. Wolf‐Gladrow (1993), Potential
vorticity constraints on the dynamics and hydrography of the Southern
Ocean, J. Phys. Oceanogr., 23, 456–487.
Marshall, J., E. Shuckburgh, H. Jones, and C. Hill (2006), Estimates and
implications of surface eddy diffusivity in the Southern Ocean derived
from tracer transport, J. Phys. Oceanogr., 36, 1806–1821.
McClean, J. L., P.‐M. Poulain, J. W. Pelton, and M. L. Maltrud (2002),
Eulerian and Lagrangian statistics from surface drifters and a high‐
resolution POP simulation in the North Aatlantic, J. Phys. Oceanogr.,
32, 2472–2491.
Nakamura, N. (1996), Two‐dimensional mixing, edge formation, and permeability in area coordinates, J. Atmos. Sci., 53, 1524–1537.
Oh, S. I., V. Zhurbas, and W. S. Park (2000), Estimating horizontal diffusivity in the East Sea (Sea of Japan) and the northwest Pacific from satellite‐tracked drifter data, J. Geophys. Res., 105, 6483–6492.
Olbers, D., and M. Visbeck (2005), A model of the zonally averaged stratification and overturning in the Southern Ocean, J. Phys. Oceanogr., 35,
1190–1205.
Phillips, H. E., and S. R. Rintoul (2000), Eddy variability and energetics
from direct current measurements in the Antarctic Circumpolar Current
south of Australia, J. Phys. Oceanogr., 30, 3050–3075.
Poulain, P.‐M., and P. P. Niiler (1989), Statistical analysis of the surface
circulation in the California Current System using satellite‐tracked drifters, J. Phys. Oceanogr., 19 (10), 1588–1603.
Poulain, P.‐M., A. Warn‐Varnas, and P. P. Niiler (1996), Near‐surface circulation of the Nordic seas as measured by Lagrangian drifters, J. Geophys.
Res., 101, 18,237–18,258.
C06006
Sallée, J. B., K. Speer, R. Morrow, and R. Lumpkin (2008), An estimate of
Lagrangian eddy statistics and diffusion in the mixed layer of the Southern
Ocean, J. Mar. Res., 66, 441–463.
Shuckburgh, E., H. Jones, J. Marshall, and C. Hill (2009a), Robustness of
effective diffusivity diagnostic in oceanic flows, J. Phys. Oceanogr., 39,
1993–2009.
Shuckburgh, E., H. Jones, J. Marshall, and C. Hill (2009b), Understanding
the regional variability of eddy diffusivity in the Pacific sector of the
Southern Ocean, J. Phys. Oceanogr., 39, 2011–2023.
Smith, K., and J. Marshall (2009), Evidence for deep eddy mixing in the
Southern Ocean, J. Phys. Oceanogr., 39, 50–69.
Speer, K., S. R. Rintoul, and B. M. Sloyan (2000), The diabatic Deacon
cell, J. Phys. Oceanogr., 30, 3212–3222.
Stammer, D. (1998), On eddy characteristics, eddy transports and mean
flow properties, J. Phys. Oceanogr., 28, 727–739.
Swenson, M. S., and P. P. Niiler (1996), Statistical analysis of the surface
circulation of the California Current, J. Geophys. Res., 101(C10),
22,631–22,645.
Taylor, G. I. (1921), Diffusion by continuous movements, Proc. R. Soc.
London A, 64, 476–490.
Treguier, A. M. (1999), Evaluating eddy mixing coefficients from eddy‐
resolving ocean models: A case study, J. Mar. Res., 57, 89–108.
Treguier, A. M., I. M. Held, and V. D. Larichev (1997), Parameterization of
quasigeostrophic eddies in primitive equation ocean models, J. Phys.
Oceanogr., 27, 567–580.
Veneziani, M., A. Griffa, A. M. Reynolds, and A. J. Mariano (2004), Oceanic
turbulence and stochastic models from subsurface Lagrangian data for the
north‐west Atlantic Ocean, J. Phys. Oceanogr., 34, 1884–1906.
Veneziani, M., A. Griffa, Z. D. Garraffo, and E. P. Chassignet (2005a),
Lagrangian spin parameter and coherent structures from trajectories
released in a high‐resolution ocean model, J. Mar. Res., 63, 753–788.
Veneziani, M., A. Griffa, A. M. Reynolds, Z. D. Garraffo, and E. P.
Chassignet (2005b), Parameterizations of lagrangian spin statistics and
particle dispersion in presence of coherent vortices, J. Mar. Res., 63,
1057–1083.
Veronis, G. (1975), The role of models in tracer studies, in Numerical Models
of Ocean Circulation, pp. 133–146, Natl. Acad. of Sci., Washington, D. C.
Visbeck, M., J. Marshall, and T. Haine (1997), Specification of eddy transfer coefficients in coarse‐resolution ocean circulation models, J. Phys.
Oceanogr., 27, 381–402.
Waugh, D. W., and E. R. Abraham (2008), Stirring in the global surface
ocean, Geophys. Res. Lett., 35, L20605, doi:10.1029/2008GL035526.
Zhurbas, V., and S. I. Oh (2003), Lateral diffusivity and Lagrangian scales
in the Pacific Ocean as derived from drifter data, J. Geophys. Res., 108
(C5), 3141, doi:10.1029/2002JC001596.
Zhurbas, V., and S. I. Oh (2004), Drifter‐derived maps of lateral diffusivity
in the Pacific and Atlantic oceans in relation to surface circulation patterns, J. Geophys. Res., 109, C05015, doi:10.1029/2003JC002241.
S. T. Gille, A. Griesel, J. L. McClean, and J. Sprintall, Scripps Institution
of Oceanography, University of California, San Diego, 8622 Discovery
Way, La Jolla, CA 92093‐0230, USA. (agriesel@ucsd.edu)
J. H. LaCasce, Meteorology and Oceanography Section, Department of
Geosciences, University of Oslo, PO Box 1022 Blindern, N‐0315 Oslo,
Norway.
M. E. Maltrud, Los Alamos National Laboratory, PO Box 1663, Los
Alamos, NM 87545, USA.
18 of 18